Cosmology Inflaton Potential Structural Inflaton Regime

The five regimes

In physical cosmology, inflation is the brief exponential expansion of the universe in its first instants. The Recognition Science (RS) framework models the potential energy driving that expansion as a single field with five canonical structural regimes. The declaration InflatonRegime names them: slow-roll plateau, slow-roll slope, hilltop decline, reheating, and the post-reheating radiation era. A theorem in the framework's machine-checked library proves the count is exactly five, and a certificate bundles that fact with the other defining constants of the model.

The framework's account of inflation is quantitative. The slow-roll parameters, which measure how flat the potential is, are set to ε = 1/(2φ⁵) and η = 1/φ⁵, where φ is the golden ratio. The spectral index n_s, which describes how density fluctuations vary with scale, is fixed by the relation n_s − 1 = −2/45, placing n_s in the band (0.955, 0.957). The total number of e-folds, the measure of how much the universe expanded, is defined as 44. The framework derives this from its gap-45 ladder, subtracting one tick for the reheating transition.

These numbers are not fitted. The framework derives the golden ratio φ from its foundational cost function, and the constants flow from that derivation. The spectral index band is proved in the library to lie within (0.955, 0.957). The slow-roll parameters are proved positive, and the identity φ⁵ = 5φ + 3 is proved as a Fibonacci relation. The certificate InflatonCert packages all of these as a single object, asserting the five regimes, the 44 e-folds, the positive parameters, and the spectral index band.

The declaration is a definitional model, not an empirical measurement. It does not claim that the actual universe's inflation followed these exact phases, nor that the spectral index has been measured to be in this band. The framework's library proves the internal consistency of the model, not its correspondence to observation. The comparison to measured cosmological data is an empirical check that the framework does not perform in this declaration.

inductive InflatonRegime where
  | slowRollPlateau
  | slowRollSlope
  | hilltopDecline
  | reheating
  | radiationEra
  deriving DecidableEq, Repr, BEq, Fintype

theorem inflatonRegime_count : Fintype.card InflatonRegime = 5 := by decide

/-- e-fold count N_e = 44 (gap-45 ladder). -/
def efoldCount : ℕ := 44

/-- n_s - 1 = -2/45 gives n_s ∈ (0.955, 0.957). -/
theorem spectralIndex_band :
    ((0.955 : ℝ) < 1 - 2/45) ∧ (1 - 2/45 < (0.957 : ℝ)) := by
  refine ⟨?_, ?_⟩ <;> norm_num

What this page does not claim

The declaration does not claim the actual universe's inflation followed these exact phases. The spectral index has not been measured to lie in the band (0.955, 0.957) by this declaration. The framework does not derive the fine-structure constant α from this model.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/InflatonPotentialStructural.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: